BERKELEY, CA — David H. Bailey,
chief technologist of the Department of Energy's National
Energy Research Scientific Computing Center (NERSC) at
Lawrence Berkeley National Laboratory, and his colleague
Richard Crandall, director of the Center for Advanced
Computation at Reed College, Portland, Oregon, have taken
a major step toward answering the age-old question of
whether the digits of pi and other math constants are
"random." Their results are reported in the
Summer 2001 issue of Experimental Mathematics.

Pi, the ubiquitous number whose first few digits are
3.14159, is irrational, which means that its digits run
on forever (by now they have been calculated to billions
of places) and never repeat in a cyclical fashion. Numbers
like pi are also thought to be "normal," which
means that their digits are random in a certain statistical
sense.

Describing the normality property, Bailey explains that
"in the familiar base 10 decimal number system, any
single digit of a normal number occurs one tenth of the
time, any two-digit combination occurs one one-hundredth
of the time, and so on. It's like throwing a fair, ten-sided
die forever and counting how often each side or combination
of sides appears."

Pi certainly seems to behave this way. In the first six
billion decimal places of pi, each of the digits from
0 through 9 shows up about six hundred million times.
Yet such results, conceivably accidental, do not prove
normality even in base 10, much less normality in other
number bases.

In fact, not a single naturally occurring math constant
has been proved normal in even one number base, to the
chagrin of mathematicians. While many constants are believed
to be normal -- including pi, the square root of 2, and
the natural logarithm of 2, often written "log(2)"
-- there are no proofs.

The determined attacks of Bailey and Crandall are beginning
to illuminate this classic problem. Their results indicate
that the normality of certain math constants is a consequence
of a plausible conjecture in the field of chaotic dynamics,
which states that sequences of a particular kind, as Bailey
puts it, "uniformly dance in the limit between 0
and 1" -- a conjecture that he and Crandall refer
to as "Hypothesis A."

"If even one particular instance of Hypothesis A
could be established," Bailey remarks, "the
consequences would be remarkable" -- for the normality
(in base 2) of pi and log(2) and many other mathematical
constants would follow.

This result derives directly from the discovery of an
ingenious formula for pi that Bailey, together with Canadian
mathematicians Peter Borwein and Simon Plouffe, found
with a computer program in 1996. Named the BBP formula
for its authors, it has the remarkable property that it
permits one to calculate an arbitrary digit in the binary
expansion of pi without needing to calculate any of the
preceding digits. Prior to 1996, mathematicians did not
believe this could be done.

The digit-calculation algorithm of the BBP formula yields
just the kind of chaotic sequences described in Hypothesis
A. Says Bailey, "These constant formulas give rise
to sequences that we conjecture are uniformly distributed
between 0 and 1 -- and if so, the constants are normal."

Bailey emphasizes that the new result he and Crandall
have obtained does not constitute a proof that pi or log(2)
is normal (since this is predicated on the unproven Hypothesis
A). "What we have done is translate a heretofore
unapproachable problem, namely the normality of pi and
other constants, to a more tractable question in the field
of chaotic processes."

He adds that "at the very least, we have shown why
the digits of pi and log(2) appear to be random: because
they are closely approximated by a type of generator associated
with the field of chaotic dynamics."

For the two mathematicians, the path to their result
has been a long one. Bailey memorized pi to more than
300 digits "as a diversion between classroom lectures"
while still a graduate student at Stanford. In 1985 he
tested NASA's new Cray-2 supercomputer by computing the
first 29 million digits of pi. The program found bugs
in the Cray-2 hardware, "much to the consternation
of Seymour Cray."

Crandall, who researches scientific applications of computation,
suggested the possible link between the digits of pi and
the theory of chaotic dynamic sequences.

While other prominent mathematicians in the field fear
that the crucial Hypothesis A may be too hard to prove,
Bailey and Crandall remain sanguine. Crandall quotes the
eminent mathematician Carl Ludwig Siegel: "One cannot
guess the real difficulties of a problem before having
solved it."

Among the numerous connections of Bailey's and Crandall's
work with other areas of research is in the field of pseudorandom
number generators, which has applications in cryptography.

"The connection to pseudorandom number generators
is likely the best route to making further progress,"
Bailey adds. "Richard and I are pursuing this angle
even as we speak."

For more about the normality of pi and other constants,
visit David
Bailey's website. The BBP algorithm for calculating
binary digits of pi was found using the PSLQ algorithm
developed by Bailey and mathematician-sculptor Helaman
Ferguson; it is discussed at Bailey's website and also
in the Fall
2000 issue of Berkeley Lab Highlights.

The Berkeley Lab is a U.S. Department of Energy national
laboratory located in Berkeley, California. It conducts
unclassified scientific research and is managed by the
University of California.